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C HAPTER 9 Combined Convection 9.1 INTRODUCTION T he buoyancy forces that arise as the result o f t he temperature differences and which cause the fluid flow in free convection also exist when there is a forced flow. The effects o f t hese buoyancy forces are, however, usually negligible when there is a forced flow. I n ~ome cases, however, these buoyancy forces d o h ave a significant influence on· t he flow and consequently on the heat transfer rate. I n s uch cases, the flow about the body is ,a combination or mixture o f forced and free convection as indicated i n Fig. 9.1 and such flows are referred to as combined o r m ixed forced and free (or natural) convection. S uch combined convective flows are normally associated with low forced velocities. They c9B ~occur, for example, i n some electronic cooling ~ituations and in some h eat exchangers. 9.2 GOVERNING PARAMETERS Consider combined convective flow over a series o f geQmetrically identical bodies as indicated i n Fig. 9.2. I n s uch a case, as discussed i n C hapter 1, the/mean heat transfer coefficient, h, will depend on: • T he conductivity, k, o f t he fluid with which the body is exchanging heat. • T he viscosity, p" o f the fluid with which the body is exchanging heat. • T he specific heat, cP ' o f t he fluid with which the body is exchanging heat. 426 CHAPrBR 9: Combined Convection 427 Forced Convection Forced Flow - Forced Flow - Free Convection --- - Combined Convection . F IGURE 9.1 Combined convective flow. t F IGURE 9.2 Flow situation considered. Direction o f ... ~ Forced Flow ~ - The density, p, of the fluid with which the body is exchanging heat. ~ - The size of the body as specified by some characteristic dimension, e. - The magnitude of the forced fluid velocity, U, relative to the body. - The buoyancy force parameter, (3 g (Tw - T f ), - The angle, </>, between the direction of the forced velocity and the direction of the gravity vector, i.e., between the direction of the forced velocity and the direction i n which the buoyancy forces act (see Fig. 9.3). 428 Introduction to Convective Heat Transfer Analysis Ut!! Flow Forced J:~ Heated Body ~~. Forced Flow tfJ = 180°, Opposing Flow Heated Body tfJ =0°, Assisting Flow FIGURE 9.3 Angle between direction of forced flow and that of the buoyancy force. I t is assumed, therefore, that: h = function[ k, IL, Cp, p, e, u, f3 g (Tw Tf), T f ), cf>] (9.1) which can b e written as: f [h, k ,IL, cp, p, e, U, f3g(Tw - cf>] =0 (9.2) where f is some function. T here are, thus, nine dimensional variables involved i n this type o f convection. As a result, there are five dimensionless variables involved i n describing combined convective heat transfer. Since the angle, cp, is dimensionless, these dimensionless variables are: '171 = '172 = '173 (he/k) = Nu, the Nusselt number = f 3g(Tw - = ( Ue/v) = Re, the Reynolds number 3 To)p 2 e /IL2 = f3g(Tw - To)e3/v 2 = G r, t he Grashofnumber ( Uep/IL) '174 = (CpIL/k) = P r, the Prandtl number '175 = cp S = Separation Point ~~ f f II J Assisting Flow Opposing Flow F IGURE 9.4 Assisting and opposing combined convective flow. CHAPI'ER 9: Combined Convection 429 Therefore, in combined convection, in general function(Nu, Re, Gr, Pr, cp) = 0 which can be rewritten as (9.3) Nu = function(Re, Gr, Pr, cp) (9.4) W hen the angle cp is 00 , the forced flow is in the same direction as the buoyancy forces. In this case, the flow is referred to as "aiding" o r "assisting" combined convective flow. When cp i s 1800 , the forced flow is in the opposite direction to the buoyancy forces. In this case the flow is referred to as "opposing" combined convective flow. The flows in these two cases can be very different particularly i f the flow involves significant regions o f separated flow. In assisting flow, the buoyancy forces tend to delay the separation, i.e., to move the separation point rearward on the body whereas in opposing flow they tend to move the separation point forward on the body. This is illustrated i n Fig. 9.4. T he effect o f the flow direction on the flow field in mixed convection is further illustrated by the experimental results given in Fig. 9.5. This figure essentially shows Assisting Opposing Horizontal Cross-Flow Forced F IGURE 9.5 Effect of flow direction on the temperature field around a heated cylinder. (From Krause, I.R., "An Interferometric Study of Mixed Convection from a Horizontal Cylinder to a Cross Flow of Air", M.E.Sc. Thesis, The University of Westem Ontario, London, Ontario, Canada\198S. By permission. ) 430 Introduction to Convective Heat Transfer Analysis the distribution of the isothenns (lines o f constant temperature) near a heated cylinder i n' a vertically upward, a vertically downward, and a horizontal air flow under such conditions th~t the buoyancy forces are important, i.e., in combined convective flow. F or comparison, the isotherms in an effectively forced convective flow are also shown. 9.3 GOVERNING EQUATIONS Attention will initially be restricted to two-dimensional steady laminar flow. T he Boussinesq assumptions will again b e used and, consistent with this assumption, dissipation effects will be neglected. The governing equations, expressed in Cartesian coordinates, are then [1]: . Continuity: (9.5) x-momentum: 2 2 au uau -+v- = -.1 -p v · _ + - +{3g(T-Tt)coscP - a + (a u a u) ax ay p ax a x2 ay2 y-momentum: ( 9.6) 2 2 av 1 ap . uav -+v- = - --+v (a v a- +{3g(T-TdsIncP -2 ay2 + v) ax ay p ay . ax Energy: (9.7) (9.8) , I n these equations, p is the pressure relative to the local ambient pressure and, as before, cfJ i s t beangle between the direction o f the forced velocity and the direction o f t he buoyancy forces as defined in Fig. 9.3. The x-axis is in the direction o f the undisturbed forced flow. T he boundary layer fonns o f the governing equations given above can be derived I . in the same way as in forced convection. The restUtant equations are: au au u ax + v ay = au + av = 0 a x ay 1 dp a2u - pdx + v ay2 + {3g(T - Tt)coscP (9.9) (9.10) aT aT u ax + v ay = (k p Cp ) a2T ay2 (9.11) / where, as i n forced convection, d p/dx i s the pressure gradient i n the free-stream w hich is imposed on the boundary l a y e r . ' . . CHAPTER 9: Combined Convection 431 A consideration of the orders of magnitude of the terms i n the momentumequation for boundary layer flow indicates that if u = O(Ul), where Ui is a characteristic free-stream velocity, then the buoyancy force term will b e important i f: G = f3 g (Twr - Tl)coscf>L ur (9.12) is of the order of magnitude, 1. Here L is some characteristic dimension of the body and Twr is a measure of the surface temperature.IfG is of a significantly lower order than one, the buoyancy force term will be negligible andforced convective flow will exist. On the other hand, if G is of 'a significantly greater order of magnitude than one, the buoyancy force effects will predominate and the flow will essentially be free convective. Hence, combined 'convective flow exists when: ' . G = 0(1) (9.13) G is usually termed the "buoyancy force parameter". I t is a'form ofR1chardson number. Now it will be noted that: G = (f3g(Twr - Tl)COScf>L3)(~) = v2 urL2 Gr R e2 (9.14) Therefore, the parameter G rl Re2 will be important i n determining, whether a boundary layer flow can be treated as a forced convective flow or as a free convective flow or as a combined convective flow. 9.4 , LAMINAR BOUNDARY LAYER F LOW O VER AN I SOTHERMAL VERTICAL FLAT P LATE' The flow situation being considered here is shown in Fig. 9.6. t U t,Tt ' x ~::d l lllll , t Buoyancy Force I f Tw> Tl t BUOyanCy Force I f Tw>Tl Assisting Flow I f Plate Heated Opposing Flow I f Plate Heated / F IGURE 9.6 Flow $ituation considered. 432 Introduction to Convective Heat Transfer Analysis Studies o f this type of flow are described in [2] to [14]. The governing equations are: au ax au uax + av ay =0 (9.15) (9.16) (9.17) + v -' au a2u = v - ± (3g(T -- Tt) ay ay2 aT, aT u ax + v ay = (k )aa2T y2 p Cp It has been noted that because flow over flat plate is being considered, the pressure gradient d pldx is O. The ( +) and (--) signs on the buoyancy t erm refer to assisting and opposing flow, respectively. IIi seeking a solution to the above equations, ~t seems worth first investigating whether similarity solutions can be o\?tained in the same way as in forced convection. Therefore, the similarity variable used in forced convection, Le.: TJ = y fU1 = !...Reo. s VVx x x (9.18) is again introduced and it is assumed that. similar velocity and temperature profiles exist in the boundary layer, i.e., that: ~ = F'(TJ), Ul T -- Tt T w--Tl = H(TJ) (9.19) The prime, of course, again denotes differentiation with respect to TJ· As'in forced convection, it follows from the continuity equation by using the boundary ct>ndition TJ ,:"" 0, ,v = 0 that: ~.= ! J Ul 2 XUl V (TJF' -- F) (9.20) Substitutiq.g this into the momentum equation and using: aTJ TJ aTJ 1 a x = - - 2x' a y = -- x /fxl vx (9.21) , gives the following equation: F 'FII(-!L)+[!J v ( F '_F),]FIIJU1X = 2x 2 ' Ul X TJ x V Ul LFIII(~)'± (3g(Tw --Tl)H xv u~ i.e.: F"' + F~" ± GxH :;: : 0 (9.22) The primes,of course, denote differentiation with respect to TI and: Gx = (3g(Tw - - Tl)X __ Grx u~ -- Re~ / CHAPTER 9: Combined Convection 433 and, as discussed before, the ( +) and ( - ) signs on the buoyancy term refer to assisting and opposing flow, respectively. Similarly, the energy equation gives as in forced convection: H " + Pr H ' F = 2 When y = 0, ° ° ° (9.24) T he boundary conditions on these equations are: U = 0, i.e., when 11 = 0, F ' = W hen y = 0, v = 0, i.e., when 11 = 0, F = W hen y is large, U ~ UI, W hen y = 0, T = T w, i.e., when 11 = 0, H = 1 i.e., when 11 is large, F ' ~ 1 W hen y i s large, T ~ (9.25) TI, i.e., when, 11 is large, H ~ ° A consideration ofEq. (9.22) shows that because Gx is a function of x, similarity solutions do not exist, i.e., F a ndH cannot be expressed as functions o f l1ruone. . To exaniine the conditions under which the buoyancy forces can be neglected, consideration is given to the case where Gx is small and the solution differs by/only a small amount from that existing in purely forced convection: I f terms o f the order o f Gx 2 a nd higher are neglected because Gx is small, the solu~on will have the form: F = Fo(l1) H= + GxFI(l1) Ho(l1) + GxHI(l1) (9.26) (9.27) • Here Fo a nd Ho are the functions that apply in purely forced convection. Substituting these equations into Eq. (9.22) then gives, since terms o f the order Gx 2 are being negl~cted: F. III + G F 'I' + FoFo' + G FoF~' + G FIFo' + G R o Xl 2 x2' x2 -xO But the analysis o f forced convective gives: .1'0 D ill. = 0 (9.28) F 2-'+ -oFo' - ° ° (9.29) Hence, Eq. (9.28) gives on canceling the Gx in all the remaining terms: F F uFIill + -oFo' + FoFl' + -IFo' ± n o 2- - 22- (9.30) This equation allows F l (11) to b efound,'the boundary conditions being: 11 = O"F} = Q 7J = 0, F l = 11 = 0, HI ! ilarge, 11 large, H I ~ ° = F} ° ° 1 ~ (9.31) / 434 Introduction to Convective Heat Transfer Analysis · Hence, using the known variation o f Fo and Ho with 'TI, E q.(9.30) c an be used to solve for F l. Similarly, substituting Eqs. (9.26) and (9.27) into Eq. (9.24) gives when terms o f the order of Gx 2 and larger are neglected: Ho' Pr + Pr x + - 2- + - 2- x + - 2- xH " F: H ' J.+~ = 0 HI' G FoHo FIHo G FoH! G - 0 (9.32) But the forced convection solution gives: Pr 2 so this equation gives: H -I' -2 +FoH! - 0 + FIHo - - Pr 2 (9.33) Because F o('TI), H ij( 'TI),. and F~ ('TI) are known, this equation can be a sed to solve for H I. The solution will depend on the value o f Pr. O nce t he values o fFI a nd H I have been determined, the heat transfer rate a t the wall can b e found b y ~pplying Fourier's law which gives: i.e.: Nux = _ HQ(O) - GxHl (0) j Rex I f the buoyancy forces are negligible, this equation gives: (9.34) NUxF = _ HQ(O) j Rex NUxF being tl).e t()cal Nusselt number in forced convection. Dividing the above two equations then gives: (9.35) Nux = 1 + Hl(O) Gx NUxF HQ(O) (9.36) where Nux is the local Nusselt number i n the actual combined convective flow. Similarly, the wall shear stress is given by: 'Tw au " = / La I =o = /LuI[Fo(0) + Gx F"(O)] I yy 'TW2 j Rex x i.e.: j Rex = Fo'(O) - GxF!'(O) (9.37) / P Ul CHAPI'BR 9: Combined Convection 435 I f the buoyancy forces are negligible, this gives: 'TwF pur J Re x = Fo'(O) 'TwF being the local wall shear stress in forced convection. Dividingtbe above two equations then gives: _ F1 '(0)G -'Tw - 1 + - - x 'TwF Fo'(O) (9.38) where 'Tw is the. actual local wall shear stress in the combined convective flow. A "shooting" technique can b e used in solving Eq. (9.30), it being assumed that the functions Fo and Ho are known. Basically, a type of iterative solution procedure is used in which the value of F1'(0) is guessed and the value of Fl at large 71 is then calculated by numerically integrating Eq. (9.30). The value of F1'(0) that gives Fl = 0 at large TJ is then iteratively determined. Once Fl(TJ) is found in this way, Eq. (9.33) can be integrated to give Hl(TJ) and the value of HI (0) can then b e determined. The solution depends on the value of the Prandtl number, Pro A computer program, SIMPLCOM, that implements this procedure in a very simple, basic manner is available as discussed in the Preface and some results given by this program are given in Fig. 9.7. A consideration of the results given in Fig. 9.7 in conjunction with Eqs. (9.35) and (9.36) shows that the buoyancy forces increase the heat transfer rate and wall shear stress in assisting flow and decrease these quantities in opposing flow. I f it • is assumed that the effect of the buoyancy forces on the heat transfer rate can b e neglected, i.e., that the flow can be assumed to be a purely forced convective flow, when: I Nux - 11 < 0.01 NUxF . 6r-----------~----------~ 4 2 Hj(O)IHo(O) o ' - - - - - _ - - - 1 1 . . . . -_ _ _ _........ 0.1 1 Pr 10 F IGURE 9.7 Variation o f HI (O)IHo(O) and Fi'(O)IFo'(O) with Prandtl number. 436 Introduction to Convective Heat Transfer Analysis i t will b e seen that, according to Eq. (9.35), purely forced convection will exist i f: 0.01 Gx < IH1(O)IHQ<O)1 E XAMPLE 9 .1. (9.39) A ir at a temperature of 20°C flows upward over a wide.verticallO-cm high flat plate which is maintained at a uniform surface temperature of 60°C. Below what velocity will the buoyancy forces have an effect on the heat transfer rate at the trailing edge o f the plate? Solution. Here, the buoyancy forces will act in the same direCtion as the forCed flow, i.e., assisting flow will exist. In this case, buoyancy force effects on the heat transfer rate ,will be important if: 0.01' Gx > IH1 (0)IHO(O)1 , ,:. For airfor which P r = 0.7, the solution discussed above gives: H{(O) _ HO(O) - 0.8 Hence, in this case, buoyancy force effects will be important if: Gx > 0.01 0.8 :i.e.: Gx > 0.0125 Now, for a vertical plate At the trailing edge, i.e., the uppermost edge of the plate, x Hence, \)uoyancy forces will be important ,at the trailing edg,e if: f3g(T w ~ Tl)O.1 > 0.0125 u1 ' = 10 em = 0.1 m; i.e., i f: 2 Ul < f3g(Tw - Tl)O.l 0 .0125, The mean air temperature in the boundary layer is: 20 + 60 2 A t this temperature: ,.. a, = 400C , =.273 1+ 40 -_ 313,K1 / 1 CHAPTER. 9: Combined Convection 437 Hence, buoyancy forces will b e important i f: 2 (1/313) X "1 < "1 9.81 X (60 - 20) X 0.1 0.0125 i.e.: < 3.17 mls Therefore, the buoyancy forces can be expected to inft.uence the heat transfer rate when the forced velocity is less than about 3.2 mls. The above analysis applies only for very small values of G x because only the first-order terms, i.e., terms of the order of Gx , were considered. The analysis could have been expanded to include some o f the hi~er order terms, e.g., Eqs. (9.26) and (9.27) could have been written as: F = Fo + GxFt + G~F2 + G~.p3 + G!F4 + GxHt + G~H2 + G~H3 + G! H 4 'and: H = 80 and the analysis could have been cEtrrled out retaining terms up to and including those involving G! but ignoring higherordet teI'IIlS in Gx • The procedure used is basically an extension of that described above to deal with the first-order analysis. However, the analysis becomes quite involved,~d i f results for larger values of Gx ~ are required it is easier to apply a numerical solution procedure of the type discussed in the next section. Before turning to a discussion of other methods o f solving the laminar boundary layer equations for combined convection, a series-type solution aimed at determining the effects 9f small forced velocities on a free convective flow will be considered. In the analysis given above to determine the effect of weak buoyancy forces on a forced flow, the similarity vari~bles for forced convection were applied to the equations for combined convection. Here, the similaritY variables that were 'previously used in obtaining a solution for free convection will be applied to these equations for combined convection. Therefore, the followirig similarity variable is introduced: , 11 = l. G rO.25 = Y [f3 g (Tw ..;.. T t ) ]0.25 X (9AO) x 112 X ' where G rx is the local Grashof number. As in free convectio~ it is assumed that: --;::::~::;===:;::::== = F '(11) u (9Al) J f3g(Tw - Tt)x and: T - Tl T - T := #(71) w I, (9A2) / The prime, of course, again denotes differentiatipn with respect to 11. Th~ differentiated function F ' is, as in forced flow, used for convenience in describing the 4 38 Introduction to Convective Heat Transfer Analysis velocity profile. The functions F' and Hare, of course, different from those in forced convective flows. ' As with purely free convective flow, the continuity equation gives: v J f3g(Tw = - Tt)x 4G~·25 1 (TJF' - 3F) (9.43) Substituting this into the momentum equation and using: dTJ = dX -!L dTJ = G~·25 x 4 x' d y (9.44) gives: F'F" ( - !L) + F,2 + 4x 2x 1 (TJF' _ 4GrQ·25 , " x 3F)F'~ G~·25 ' x, =-±~ , Fill , H x x i.e.: Fill + 3 F:" _ F~2 ±H = 0 (9.45) Similarly, the energy equation.gives: F 'H' ( - ~ ) + 4G~25 ('1F'. , H" 3F)H' G ;25 H" , P rx i.e.: + ~PrFH' 4 = 0' (9.46) , A consideration of Eqs.' (9.45) and (9.46) might, first sight, appear to indicate'that a similarity solution can be obtained in this case. However, consider'the boundary conditions: at = 0: u, = 0, i.e., at TJ = 0: F ' = 0 At Y = 0: v = 0, i.e., at TJ = 0: F = 0 At Y = 0: T == T w, i.e., at TJ = 0: H = 1 At y For large y: u ~ For large y: T i t having been noted that: for large TJ: F ' ~ [f3g(Tw Ut - (9.47) U t. i.e., for large TJ: F ' ~ 0;°·5 ~ T t. i.e., for large TJ: H ~ 0 Tt)xt 5 = G~·5 = G~5 Rex 1 The 'solution thus depends on Gx so a similarity solution cannot be obtained. As Gx tends to infinity, the :flow tends towards a purely free convective flow./ To examine the conditions under which the 'effects ,of the forced velocitY,: on it free convective flow can be neglected, a series solutiori in t ems o f l IGx 0.5, 'will be con- CHAPTER 9: Combined Convection 439 s idered a nd t erms o f t he order o f ( lIG x O. S )2 and higher will be neglected, i.e., i t w ill b e a ssumed that: (9.48) . (9.49) I n t his case, Fo a nd Ho are the functions that apply i n p urely free convection. Substituting these equations into Eq. (9.45) and ignoring terms involving (lIG~·S)2 t hen gives: (9.50) O nly t he ( +) sign has been accounted for o n t he buoyancy term, i.e., only the case w here t he forced flow is i n t he same direction as t he b uoyancy forces has b een considered. This is because in opposing flow the outer forced flow would b e i n t he opposite direction to the buoyancy-driven flow near the surface. I n this case the bound~ ary l ayer equations would not apply. I t follows from the analysis o f free convection that: 3D F, III + - ro F," 4 ° F u° - -2fl' + no- 0 (9.51) so, Eq. ( 9.50) gives: 3D 3 FIill + - ro FI" + -FIFo' ~ F,'F' + H I - 0 . I .4 4 - -. 2 ° (9,,52) T he b oundary conditions give: 'Tl 'Tl = 0, Fo = 0, Fl = 0 = 0, Fo = 0, F l = 0 (9.53) 'Tl = 0, Ho = 1, H I\= 0 'Tllarge, Fa ~ 0, F:{ . ,-+ 1 'Tllarge, Ho ~ 0, H I ~ 0 T he b oundary condition on F ' a t large 'Tl i s obtained b y noting that: ' F{ 1 'Tl Iarge, Fo + GO.s ~ GO.s x x B ut t he f ree convective flow solution requires that Fo ~ 0 a t l arge 'Tl so the above equation requires that: ''Tllarge, F{ ~1 440 Introduction to Convective Heat Transfer Analysis Next, Eqs. (9.48) and (9.49) are substituted into the energy equation, i.e., Eq. (9.46). This gives to first-order accuracy in I/G~·5: H" HI' ' 3 3 ° + GO.5 + "34 PrFoH0+ 4G0.s PrFoHI' + 4GO.5 P rF laO = 0 x x x T T' (9 54) . B ut the free convective flow solution requires: 3 , H " + "4PrFoHo so Eq. (9.54) gives: ° = 0 (9.55) (9.56) Therefore, since the purely free convection solution gives F0 a nd Ho, Eqs. (9.52) and (9.56) can b e simultaneously solved to give F I a nd H I u sing the boundary conditions given i n Eq. (9.$3). The heat transfer rate is then given b y applying Fourier's law at the wall. This gives: a q w-- k· - T ay IGrQ.25 N ux x y=O. - - k(T·w- T I ) [H'(O)· + - -G- 0 Hl(O)] r~·25 G~·5 X i.e.: = - H'(O) _ H'(O)G0.5 0 I x (9.57) I f the effects o f the forced velocity are negligible, i.e., i f p urely free convection can b e assumed, Eq. (9.57) gives: Gf!.25 x N UxN= - H'(O) 0 (9.58) N UxN being the local Nusselt number in purely free o r natural convection. 10 1 Mixed Convection - 0.1 I- 0.01 0.1 I::======:c:=====:l \1, _ 1 Pr 10 F IGURE 9.8 Limiting values of Gx for purely forced and purely free convection. CHAPTER 9: ,Combined Conv,ection 441 Dividing Eq. (9.57) by Eq. (9.58) then gives: Nux = 1 + Hl<O) _1_' NUxN HQ(O) G~.5 This allows the value ,of Gx above which: ' ·,1 N ux I/VUxN -11 <0.01 ' i.e;, above which the Nusselt number is within 1% o f t he purely free conveCtive value, to be found. T hiswill b e given b y : ' .' ' Gx > I 04 1H1O)IHO<O) 12 ( (9.59) T he value o f Ox so found will depend on the value o f Pr: Some'typical vai~es are shown in Fig. 9.8 which also shows the values o f Gx below which the flow can be assumed to be purely forced ,convective. These values are given by Eq. (9.39). E XAMPLE 9.~. Consider free convective flow over a wide vertical plate, which is held at a uniiOlm surface temperature of 60°C placed in aii at a temperature of 20°C. I f a forced flow is introduced over the plate, at what air velocity will the forced flow start to affect the heat transfer rate at a distance of20 cm from the leading edge of the plate? the forced flow will be important approximately when: Gx < 12 i.e., when: f 3g(Tw ; Solution. Using the results given in Fig. 9.8, it will be seen that for air, i.e., for Pr = 0.7, " T I)x < 12 ' u1 i.e., when: 2 > f3g(Tw ~ T r)x UI " 12 ' The mean temperature in the boundary layer is (60 + 20)/2 point being consi,dered is 0.2 m. Hence, since: f3 - = 40°C and X at the , - 273 + 40 - 313 1 _ 1 K-1 it follows that forced flow effects will be important when: 2 ul > (1/313) X 9.81 X (60 - 20) X 0.2 12 " i.e., when: UI > 0.12 mls Therefore, the forced flow will affect the heat transfer at the point considered if the / forced velocity is greater thaD. 0.12 mls. ' 442 Introduction to Convective Heat Transfer Analysis 9.S NUMERICAL SOLUTION O F BOUNDARY LAYER EQUATIONS T he numerical procedure for solving the laminar boundary layer equations for forced convection that was described in Chapter 3 is easily extended to deal with combined convection. The details o f the procedure are basically the same as those for forced convection and the details will not be repeated here [16]. A computer program, L AMB MIX, based on the procedure is available in the way discussed i n the Preface. This program can actually allow the wall temperature or wall heat. flux to v ary w ith X but as available, the program i s set for the case o f a uniform ,wall ~inperature o r a uniform wall heat flux. ' . .' T his program allows the variation o f Nuxl J Re x with Gx to be found in the uniform surface temperature case and the variation o f Nuxl J Re x with: . G* = x kv2Re5/2 x f3 gQw:0 =, Re5/2 x . Gr; (9.60) to b e obtained in·the uniform heat flux case. Here Gr; is the heat fluX Grashofhumber, i.e.: . (9.61) - --,- Pure forced convection - --- Pure free convection 0.1 t o-I '--L-L..LI.LIWL...::....a....r...uu.w.L-L.L.L.I.IJWL.....I....&...LJUWiL-.&..I...LLI.&UI 1 0-2 t o-I 100 tO l 1()2 103 .' GrJRe; F IGURE 9.9 . .. Variations of N uxlJRe x with Gx for mix~ convective ,flow over a vertical plate with a uniform surface temperat,ure. CHAPrER 9: Combined Convection 443 T he variations o f N uxlJRe x with Gx for the uniform surface temperature case and o f N uxlJRe x with G~ for the uniform wall heat flux case for various values o f P r as given by the computer program are shown in Figs. 9.9 a nd 9.10, respectively. O f course, i f attention was to have been restricted to the case where the wall temperature or the wall heat flux was uniform, it would have been better to write the governing equations i n terms of dimensionless variables that directly gave the variations shown in Figs. 9.9 and 9.10, a nd not to have used the same dimensionless variables as.in forced convection. However, with the variables used, the computer program based on this procedure can easily be modified to allow for a variable wall temperature or a variable wall heat flux. I f i t i s again assumed that forced convection effectively exists when: .[. Nux NUxF -1[ < 0.01 a nd that free convection effectively exists when: [ Nux _ NUxN 1[. < 0.01 . _ • ..,.... Pure forced convection - --- Pure free convection P r= 100 - :-- Assisting flow - - - - Opposing flow l~l~~~~~~~W-~LUUW~~~WL~-U~~ 10-2 10-1 100 s12 Gr*/Rex x 102 103 / FIGURE 9.10 .. Variations of NuxlJRe x ~th a; for mixed convective flow over a vertical plate with a uniform surface heat flux. i 444 Introduction to Convective Heat Transfer Analysis 100 ~ 10 Mixed Convection 0.1 I- 0.01 0.1 1 10 . Pr FIGURE 9.11 Values of Gx for effectively purely , forced and purely free convection . 100 for a plate with a uniform surface. temperature. l00~------------~----------~ 10 Gx * 1 Mixed Convection 0.1 F IGURE 9.12 .. Values of G ; for effectively purely 0.01 " -=;;;....._--1----____L......_ ______....J forced and purely free convection 0.1 1 10 100 for a plate with a unifOrm surface Pr heat flux. then the results given in Figs. 9.9 and 9.10 can be used to deduce the variations of Gx with P r for the uniform wall temperature case for which effectively. purely forced convection and purely free convection exist and of G ; with P r for the uniform wall heat flux case for which effectively purely forced .c;onvection and purely free convection exist. These variations are shown in Figs. 9.11 and 9.12. E XAMPLE 9 .3. Air at a temperature of 15°C flows upward over a O.2-m/high vertical plate which is kept at a uniform surface temperature of 45°C. Plot the variation of the local heaftransfer nite along the plate for air velocities of 2 and 0.3 mls. Assume twodimensional flow. Solution. The mean temperature of the air in the boundary layer is (IS -+ 45)/2, At this temperature for air at standard ambient pressure: , ". r '. = 30°C. cHAPrER 9: Combined Convection 445 Now: Gx i.e.: = f3g(Tw - T1)x u2 1 = (11303) x 9.81 x 30 X x u2 1 = 0.9713 ~ U2 1 where x is in m and UI is in mls. For any value of Gx and for either value of Ul that is to be considered, this equation allows the 'corresponding value of x to b e determined. For Ul = 2 mis, this equation gives: x while for Ul = 4.12Gx ' =0.3 mls i t gives: x = 0.0927Gx Because the maximum value that x can have is 0.2 m, this shows that the highest value of Gx is 0.0486 for Ul = 2 mls and 2.158 for Ul = 0.3 mls. For any value of Gx , the numerical solution discussed above gives the corresponding • value of NuxIRe~·s. But: Re~'s i.e.: Nux = qwxlk(Tw - Tl) qwxl30 x 0.02638 (ulxlll)O.S = (UlxI0.OOOOI601)o.s qwxo.s = 197.8uY.s 300 - :W/m2 200 l00------------~ _ _________~ 0,2 0.0 0.1 X -in F lGUUE9.3 /' 4 46 Introduction to Convective Heat Transfer Analysis Hence, since for any value o f Gx the numerical solution discussed above gives the corresponding value o f Nux/Re~·5 and since, for any value o f G x and for either value o f U l t hat is to be considered, the corresponding value o f x c an be determined, the variation o f qw with x for either value o f U l that is to b e considered can be determined. For example, the numerical solution gives for Gx = 0.5,Nux/Re~·5 = 0.430. I f the case o f U l = 0.3 mls is considered, this value o f Gx c orrespondstoa value o f x that is given by: x = 0.0927G x = 0.0927 X 0.5 = 0.0464 m T he corresponding value o f qw is given by: qw = 197.8 X 0.3°·5 0 430 = 503 W 0.04640.5 X . . l ]sing this procedure, the variation o f qw with x for the two velocities considered can b e found, the results being shown in Fig. E9.3. T he numerical procedure described abo\':e for solving the laminar boundary layer equations is easily extended to deal with situations in which the free-stream velocity is varying with x, i.e., to deal with situations involving flow over bodies o f arbitrary shape. 9.6 COMBINED CONVECTION O VER A HORIZONTAL PLATE In combined convective flow over a horizontal flat surface, the buoyancy forces are at right angles to t he flow direction and lead to pressure changes across the boundary layer, i.e., there is an induced pressure gradient in the boundary layer despite the fact that flow over a flat plate is involved. Under some circumstances, this can lead to complex three-dimensional flow in the boundary layer. This type o f flow will not be considered here, more information being available in [17] to [23]. 9.7 SOLUTIONS TO T HE FULL GOVERNING EQUATIONS T he boundary layer equations can, as previously discussed, only b e applied to flows i n which the Reynolds number is relatively large and in which there is no significant areas o f reversed flow. This, in particular, severely limits the applicability o f these equations i n situations involving opposing flow. W hen these conditions are not satisfied, the solution must be obtained using the full governing equations. For example, i f t he flow can be assumed to be two-dimensional and i f the Boussinesq approximations are applicable, the equations governing the flow are Eqs. (9.5) to (9.7). I f the x-axis is vertical, these equations become: au + av = 0 ax ay 22 u 'au 1 ap ua-+v- = - --+11 (a +-.- ±/3g(T-Tl) - u a u) ax ay p ax ax2 ay2 (9.62) (9.63) CHAPTER 9: Combined Convection 447 uaV + v aV = _! aP + a a x y p ay u aT + v aT ax ay = (a 2 + a2 v v) ax2 ay2 (pk )(a2T + a2T) Cp, ax2 ay2 1/ (9.64) (9.65) T he ± sign on the buoyancy force term i n Eq. (9.64) arises because the x-axis is i n t he direction o f the forced flow and can thus b e e ither vertically upwards or vertically downwards, i.e., either i n the same direction as the buoyancy forces or i n the opposite direction to the buoyancy forces. T he above equations can b e solved using numerical methods, i.e., using the same basic procedures as used with forced convection. There is, however, one major difference between the procedures used in forced convection and i n mixed convection. In forced convection, the velocity field is independent o f t he temperature field because fluid properties are here being assumed constant. Thus, in forced convection i t is possible to first solve for the momentum and continuity equations and then, once this solution is obtained, to solve for the temperature distribution in the flow. However, i n combined convection, because of the presence o f t he temperature-dependent buoyancy force term in the momentum equation, all o f t he equations must be solved simultaneously. Studies o f flows for which the boundary layer equations are not applicable are described in [24] to [43]. To illustrate the form o f results obtained, consider assisting and opposing combined convective flow over a square cylinder at low Reynolds numbers (Fig. 9.13). A computer program, MIXSQCYL, t hat finds this solution can be obtained i n the way discussed in the Preface. The program assumes that the flow over the cylinder is symmetrical about the vertical center-line o f the cylinder. Some results obtained using this program are shown in Figs. 9.14 and 9.15. Figure 9.14 shows some typical streamline patterns. Because the flow is assumed to be symmetrical, only half o f the patterns are shown in these figures. I t will b e seen from Fig. 9.14 that, as discussed earlier i n this chapter, the buoyancy forces tend to reduce the size o f the separation region downstream o f the cylinder i n assisting flow whereas they tend to increase the size o f this separation region in opposing flow. Figure 9.15 Center L ine' x y ~ Body at Uniform Surface Temperature, Tw !" I ttttttt ::~dUl'Tl F IGURE 9.13 Flow situation considered. 448 Introduction to.Convective Heat Transfer Analysis Opposing Forced Assisting F IGURE 9.14 Typical streamline 'patterns for combined c.onvective flow over a square cylinder at a Reynolds number of 50 and a Prandtl number of 0.7. 5r-----------~------------~ G r= 2500 4 Nu 3 - 2 1 ' ----.-------'----------.....1 1 10 Re 100 F IGURE 9.15 Variation of Nusselt number with Reynolds number in forced convection and in assisting and opposing convection for a Grashof number of 2500. shows some typical changes in the Nusselt number produced by the buoyancy forces in mixed convection. As with flow over a flat plate, the buoyancy forces increase the Nusselt number in assisting flow and decrease it in opposing flow. Th,ese Nusselt / number changes are much less with flow over a square cylinder than -they are for flow over acircular cylinder because, in the latter case, the buoyancy forces cause large movements in the point of separation. These changes do not occur with a square cylinder due to the presence of the sharp comers. This is shown schematically in Fig. 9.16. CHAPI'ER ,9: Combined Convection 449 S = Separation Point F IGURE 9.16 Opposing Flow Forced Flow Assisting Flow Effect o f buoyancy forces on flow , , o ver'a square and a circular cylinder. 9.8 C ORRELATION O F HEAT TRANSFER RESULTS F OR M IXED CONVECTION Consider assisting combined convective flow, over a body. I f the Gi'ashof number is kept constant and the Reynolds number is varied, t he' variation o f Nu with Re would resemble that shown in Fig. 9.17. This type o f result could be obtained by considering a body o f fixed 'size, kept 'at a fixed temperature (this would keep the Grashof numbercons~t), that is placed in a fluid flow i n which the velocity could be varied (this would allow t he Reynolds number to b e varied). , A t 10w'Reynolds numbers~ the'Nusselt number will tend to the'Gonstant value 'that would exist in purely free cDnvection, this being designated as N UN, whereas at high Reynolds numbers, when the effects o f the buoyancy forces are small, the Nusselt numbers will tend to the values that would exist i n purely forced convection at the same Reynolds number as that being considered. These forced convection Nusselt numbers are here designated as N UF. I n the combined convection regions between these two limits, the Nusselt number\variation can be approximately GrConstant Nu F IGURE 9.17 Typical variation o f Nusselt number with Reynolds number i n assisting combined convective flow. / Re 450 Introduction to Convective Heat Transfer Analysis represented by: Nun = NUN + Nu'F (9.66) where the value o f t he index n depends on the geometrical situation being considered. For example, consider flow over a vertical isothermal plate. T he N usselt numbers i Ii the limiting purely forced and purely free convective flow cases are ,given by: (9.67) and: NUN = CNGr~·25 (9.68) T he values o f t he coefficients, CF a nd CN, are dependent on the Prandtl number o f the fluid being considered. For a Prandtl number o f 0 .7, t hey are, as discussed i n Chapters 3 and 8, given by: For P r = 0.7: CF = 0.29, eN = 0 .36 (9.69) . Substituting Eqs. (9.67) and (9.68) into (9.66) a nd d ividing through by Re~·5 gives: (9.70) . T he results given b y this equation with n s et equal to 3.5 are compared in Fig. 9.18 with the numerically calculated results for assisting flow for a Prandtl number o f 0 .7 t hat were given earlier i n Fig. 9.9. It w ill b e s een that Eq. (9.70) does, i n fact, describe t he variation in the mixed convection region with assisting flow to an accuracy that is quite acceptable for most purposes.Eq. (9.66) does, therefore, apply to assisting mixed convective flow o ver' a flat plate. I t h as beenishQwn that i t does, i n fact, also describe experimental results for other more complex situations to a good degree o f a ccuracy provided the value lr---------~----------~----------_; o Correlation Equation Numerical 0.1 '--_~___. ...I-_ _ _ _ _ _- -L._ _ _ _ _ _- --I 0.01 0.1 1 10 F IGURE 9.18 Comparison with variation of NuxIRe~'s with G x given by Eq. (9.70) with actual calculated. . variation. CHAPIER 9: Combined Convection 451 T ABLE 9 .1 Values of n for various geometries Situation Vertical plate, assisting flow Horizontal cylinder, assisting flow, uniform surface temperature Horizontal cylinder, assisting flow, uniform surface heat flux Horizontal cylinder, horizontal cross flow Sphere, assisting flow, uniform surface temperature Index, n 3.5 3.5 4 7 3.5 of n is properly chosen. Some values of n deduced' from experimental results for various geometrical situations are shown in Table 9.1, see [44] to [54]. Cross flow involves a horizontal forced flow which is thus at right angles to the buoyancy forces. The conditions under which the heat transfer rate can b e assumed to be equal to that in purely forced conv~tion and under which it can be assumed to be equal to ,that in purely free convective can be deduced from Eq. (9.66)., For this 'purpose, this equation can be written as: :; = [(~:;)" + r (9.71) I f it is again assumed that the flow can be taken to be forced convective i f: N UF - Nu -1 < 0.01 then it will be seen that Eq. (9.66) indicates that forced convection can be assumed to exist i f: " (9.72) i.e., if: ,(9.73) Thi~ defines the conditions under which the flow can be assumed to be purely forced convective. Similarly, if ~t is again assumed that the flow can be taken to b e free convective i f: HUN ~.,--1 Nu ' < 0.01 then it will beseen~at since Eq. (9.66) indicates that: Nu N UN ,~[(NUF)n . .: . 1] lin, " NUN (9.74) / 452 Introduction to Convective Heat Transfer Analysis l ~ ~ ~ 2.50 2.00 ~ " "r- . /, : -" o~ 4.~ 1.50 " " ,. ~,,~o .... ~ "~#o " " 01.00 Free Convection 0.50 . , 'Opposing Flow Oiieatation 2 ;• , ,1.50,'2.00, Nut/Nun o 0.50 , 01.00 2.50 F IGURE 9.19 • • '. ~.' .. \ r .~ " Correlation of results for opposing flow over form. of cylinder indicated [47]~ • free convection can be assumed to e4ist if: [(Z:~J -1rn < 1.01 i.e., if: NUF. < (1.01 n - ' l )ljn NUN (9.75) The above discussion applied only to assisting flow and cross flow. Now, it will . .. be noted that Eq. (9.74) indicates that:' NUN N u, .' - - ' = .lc; unctlOn (,NUcF) ' - ', - NUN (9.76) I , · Attempts have been made t o correlate results 'for other geometricaIsituation~ by assuming that this type o f relation applies. Some success has been achieved by using this approach. For example, some experimental results for opposing flow over a diamond-shaped cylinder are correlated iIi this .way in Fig. 9.19. E XAMPLE 9 .4. Air at a temperature of, 15°C~ows,.over a ~5-inm diameter horizontal cylinder which has a uniform surface temperature of 45°C. I I the air flow is also horizontal and at right angles to the axis of the cylinder, determine how the heat transfer rate per meter length from the cylinder varies with air velocity. Determine the air velocity above which the flow can be assumed to be purely forced convective. / CHAPr.ElR 9: Combined Convection 453 Solution. T he mean temperature o f the air i n the flow about the cylinder is: (15 4 5)/2 = 30°C. A t this temperature for air at standard ambient pressure: + Because cross flow over a horizontal cylinder is being considered, the results given i n Table 9.1 indicate that: n =7 i.e., that: Hence: The Grashof number for the situation being considered is given by: G = r f3g(Tw - Tl)D3 . v2 = (11303) X 9.81 x (45 - 15) X 0.025 3 (16.01 X 10-:6)2 = 5 9210 ' F or this value o f the Grashof number, standard correlation equations for natural convection from a horizontal cylinder- indicate that for a Ptandtl number o f 0.7: . .. . N UN = 5.98 Hence: Nu7 = 5.987 + Nu~ = = 2 73470+ Nu~ T he Reynolds number for the situation being considered is given by: Re = VD v O. 025 V 16.01 X 1 0-6 = 1562V where the velocity, V, is in mls. Thus, for any value o f V, the Reynolds number can be found. For any value o f the Reynolds number and for a Prandtl n umber o f 0.7; standard correlation equations for forced convection from a cyij.nder allow N UF to b e found which then, using the above equation, allows Nu to b e found. But: from which it follows that: qw = Nuk(Tw - Tl) D = Nu X 0.02638 X (45 - 15) = 3 166u 0.025 . ~yU WI 2 m T he h eat transfer rate p er m length o f the cylinder is given by: Q = qw7TD = 31.66 X Nu X 7T X 0.025 ~ 2.487Nu W 454 Introduction to Convective Heat Trans(er Analysis Therefore, for any value o f t he velocity, V, the corresponding, value o f Q c an b e found. S ome values o f Q obtained in this way are shown in t he following table. I f the effect o f t he buoyancy forces can b e neglected, Q is given by: Values o f QF are also shown i n t he table. Alternatively, i f forced flow effects are negligible, Q i s given by: QN = 2.487NuN = 2.487 x 5.98 = 14.87 W V -mls 0 0.02 0.05 0.1 0.15 0.2 0.4 0.6 0.8 1..0 1.5 Re NUF Nu Q F-W Q -W 14.87 14.89 15.18 16.95 19.57 ' 22.18 30.80 37.53 43.21 46.26 58.89 0 0 3.000 31.24 4.569 78.100 '156.2 6.337 7.693 234.3 8.837 312.4 12.37 624.8 15.09 937.2 17.37 1249.6 1562.0 " 18.60 23.68 2343.0 0 5.98 7.460 5.9875' 11.36 6.102 15.76 6.816 7.869 ' 19.13 21.98 9.917 30.77 12.38 37.52 15.09 43.21 17.38 46.26 18.60 58.89 23.68 T he h eat transfer rates are shown plotted against the velocity in Fig. E9.4., I t will be seen from the results given i n the above table that the heat transfer rate is w ithin-l% o f its forced convective value when the velocity is greater than approximately 0.2 m ls.' 50 r-----~----~------------~ 40 Purely Free Convection 10 Purely Forced : Convection f • 0 '-----,-----1.---'--------' 0.0 0.5 Velocity, V -m/s 1.0 FIGURE E9.4 CHAPTER 9: Combined Convection 455 9.9 E FFECT O F BUOYANCY F ORCES O N TURBULENT F LOWS I n a turbulent flow, the buoyancy forces have two effects, [55] to [60]: i. T he m omentum equation must be modified to include the buoyancy force. ii. T he turbulence model employed must be modified to account for the effect o f the buoyancy forces on the turbulence quantities. Consider, for example, assisting turbulent mixed convection over a vertical flat plate. T his situation is schematically shown in Fig. 9.20. I f i t i s assumed that the boundary layer assumptions apply, the governing equations for the mean velocity components and temperature are: a u+av=o ax oy (9.77) Tl) . u OU -+ v au ax ay = ~ [(V + E) ou] + I3g(T - ay ay (9.78) aT aT u OX + v ay = a [. OT] ay (ex + EH) oy (9.79) T he l ast t eon i n the momentum equation, i.e., Eq. (9.78), represents the aff~ct o f t he buoyancy forces on the mean momentum balance. However, these buoyancy forces also affect the variation o f E and E H i n the flow. To illustrate how the buoyancy forces can effect E and E H, consider again the simple mixing length model discussed i n Chapter 5. " Lumps" or " eddies" o f fluid are assumed to move across the flow through a transverse distance, em, while retaining their initial velocity and temperature. They then interact with the local fluid layer giving rise to the fluctuations i n velocity and temperature that occur i n turbulent flow. Thus, to recap the derivation, consider the situation shown i n Fig. 9.21. As discussed i n C hapter 5, i t is assumed that " lumps" from layer 2 arrive at layer 1 with a velocity defici,tof: Turbulent Boundary Layer tBUOyanCy IForces ftttt ~~~:::~'FIGURE 9.20 . = Temperature Tl / Flow situation being consider~d. 456 Introduction to Convective Heat Transfer Analysis y F IGURE 9.21 Mixing length model. while lumps from layer 3 arrive at layer 1 with a velocity excess of: a U3 = .em ay au I t is then asswi1ed that the magnitude of the velocity fluctuation at 1 is propor.tional to the average of lau21and laU31, i.e.: lu'l e x! [lau21 + laU31] Le.: lu'l ex em I:~ I I t is further assumdd that the tnmsverse velocity fluctuations arise from the lon- gitudinal velocity fluctuations in order to satisfy continuity requirements, i.e.:' Iv'l ex lu'l Lastly, it is assumed that turbulent stress is proportional to Iv'llu'l, i.e., that: 7']' =- pv' U ' ex plv'II~'1 Hence: 7']' . = au au pcue21 -l m ay ay 0a a P{' 2 1 - ul - U Le.: _ 7']' - ay ay (9.80) where Cu is a constant of proportionality that has been combined with em to give e. In writing Eq. (9.80), account has been taken of the fact that the sign of 7']' depends on the sign of aulay. Hence, laulayl has been used instead of (au/ay)2.' CH.AP'l"ER 9: Combined Convection 457 Next, consider the temperature fluctuations. The lumps arriving at. I from 3 and arriving at 1 from 2 do so with a temperature excess and deficit of: flT3 and = em ay aT respectively. As with the velocity fluctuation, it is assumed that the magnitude of the temperature fluctuation at I is proportional to the average of /L\T21 and IflT3/, Le.: IT'I oc .! [/flT21 + IflT3/]· i.e.: IT'I oc em ay aT (9.81) I t is also assumed that the turbulent heat transfer rate is proportional to Iv'IIT'I, i.e., that: .. Hence: i.e.: qT = pcpce21 ~; I ~~ (9.82) wh~re CT and C are constants of proportionality. In writing Eq. (9~82), account has beeri taken of t hi fact that the sign of qT depends on the sign of aTlay. . In the derivation of Eqs. (9.80) and (9.82), i rwas assumed that ·the buoyancy forces had no effect on the flow. However, by assumption, as the fluid "lumps" move across the flow from 2 to I and from 3 to 1 they are at a different temperature than the surrounding fluid arid, as a result, they are acted on by buoyancy forces. For example, consider a fluid "lump" moving from 2 to 1. I fa1i/ay is taken aspt)sitlve and DT/ay is taken as negative, then as the "lump" moves from 2 to 1 the temperature difference between the "lump" and the surrounding fluid will increase. When the "lump" is at 2, it is at the same temperature as th~ surrounding fluid and there is no buoyancy force / acting on it. When the lump arrives at 1 it is at a temperature that is f m(aTlay) above that of the surrounding fluid and the buoyancy force acting on it per unit volume is -f3gem(aT/ay). The average buoyancy force per unit volume acting on the lump between 2 and 1 is, therefore: 1 . . aT I / . --f3gpem . 2 ay 458 Introduction to Convective Heat Transfer Analysis This buoyancy force will increase the velocity, of the "lump" of fluid in the direction of the buoyancy force, i.e., in the x-direction, by an amount a UB that is given by Newton's law as: p at a UB = -2~gpem 1 aT ay because a unit volume is being considered. at is the time taken by the lump to move from 2 to 1. Hence: (9.83) But, because the "lump" of flUid moves a distance, em, in going in the y-direction from 2 to 1 and because the movement of the "lump" is caused by the presence of the transverse velocity fluctuation v', it follows that: A ut = lemf V' ~ Substituting this into Eq. (9.83) then gives: a UB = -2~glv'l 1 aT ay lu'l, i.e., that: (9.84) , But it has been assumed that Iv'l is proportional to Iv'l Hence,Eq. (9.84) gives: , a UB = Klu'l ~ = -2~g Klu'l ay 1 aT (9.85) Now, the "lump" of fluid from 2 will arrive at 1 with a velocity deficit (as mentioned before, au{uy is taken as positive) that is given by:, a U2 = , au em - a 'y aUB (9,.86) the buoyancy forces having increased the velocity, of the "lump", thus causing it'to anive at 1 with a reduced velocity deficit. . Substituting Eq. (9.85) into Eq. (9.86) then gives: .a U2' = au~g~ aT em ay + 2Klu'l ay (9.87) , Similarly, a "lump" of fluid moving across the flow from 3 to 1 will arrive with a velocity excess of: ', ,A U U3 = e au f3ge2amT may + 2Klu'l ay (9.88) CHAPTER 9: Combined Convection 459 I n this case, the fluid "lump" moving from 3 to 1 will have a IQwer temperature than the surrounding fluid so the buoyancy forces will reduce the veI()ciiy component in the x-direction and thus reduce the velocity difference between that of the "lump" and that of the surrounding fluid when the lump arriv~s at 1. It is now again assumed that: which gives: a Iu'I ex 'tm aU + J3gem aT y 2Klul ay 0 " 2 - (9.89) I t w ill be assumed that the effect of the buoyancy forces on the turbulence structure is relatively small, i.e., that the magnitude of the second term in Eq. (9.89) is much smaller than the magnitude of the first. Now Eq, (9.89) can b e written as: . , a u [ f3g aTla y .] lu I ex em ay 1 + 2K aU/aylaulayl But, as discussed above, it is also assumed that: (9.90) Iv'l ex lu'l hence: au 3g ---,-a---;TI--..;ay'-:---r] Iv'I ex't m . - . + -f2K- -aulaylaulayl ay 0 [1 (9.91) . It is also,as before, assumed that: 'IT = - pv'u' ex plv'llu'l _ 2 from which it follows that: . 'IT ex 21 aul a u[ . f3g aTlay ] pem ay ay 1 + 2K auta;ylaUlayl (9.92) But; as discussed above, the buoyancy force effect is assumed to be small, i.e., the second term in the bracket is assumed to be much less than 1. Hence, Eq. (9.92) can be written as: 'IT - P _ e21 ay ay [ I+ f3 g autaylaulayl ul au J a rJaY] a J=~ (9.93) where: 1 K 460 Introduction to Convective Heat Transfer Analysis Next, consider the turbulent heat transfer. The buoyancy forces have no effect on the temperature fluctuations so Eq. (9.81) still applies. Hence, again assuming: q T= - pcpv'T' oc plv'IIT'1 and using Eq. (9.91), the following is obtained: qT oc pCpem ay ay 21 aul aT [ f3g 1 + 2K aT/ay ] au/aylau/ayl i.e.: qT = 21 aul aT [ Jf3g aT/ay ] pCpce ay ay 1 + 2 au/aylau/ayl . e21 au I[1 + J{3g OuIaylau/ayl ] aT/a y ay (9.94) Eqs. (9.93) and (9.94) give: E = (9.95): and: EH = 21 aul [ J{3g aT/ay ] ce ay 1 + - 2- au/aylau/ayl (9.96) Hence: pEl rT = EH = C [1 + "J2 f3g aulaylaulayl ] aT/a y (9.97) A comparison of results given by using these equations with experiment indicates that J is approximately equal to 2. The effect of the buoyancy forces on the turbulence therefore depends on the value of the dimensionless quantity: S_ - {3g aT/ay aulaylaulayl (9.98) which is a form of Richardson number. I t will be noted that the above derivation was based on the assumpti.on that aT/ay was negative and that au/ay was positive and that the buoyancy forces acted in the x-direction, i.e., they applied to assisting flow. In opposing flow, the buoyancy forces act in the opposite direction to the x-axis, this x-axis conventionally being taken in the directiol1l of the forced flow. Thus, in general, Eqs. (9.95), (9.96), and (9.97) can . be written as: . E= 2 e 1au 1[1 ± JS] ay (9.99) (9.100) EH = ce2laul[1 ± "JS ay 2] CHAPrER 9: Combined Convection 461 (9.101) where the upper sign on J applies in assisting flow and the lower sign on J applies in opposing flow. B ecause, for flow over a heated surface, auliJy is positive and aTlay is negative, S will normally b e a negative. Hence, i n assisting flow, the buoyancy forces will tend to decrease E and E H , i.e., to d amp the turbulence, and thus to decrease the heat transfer rate below the purely forced convective flow value. However, the buoyancy force i n t he momentum equation tends to increase the mean velocity and, therefore, to i ncrease the heat transfer rate. I n turbulent assisting flow ·over a flat plate, this can lead to a Nusselt number variation with Reynolds number that resembles that shown i n Fig. 9.22. A t h igher Reynolds numbers, i.e., between points A and B in Fig. 9.22, the effect o f t he buoyancy forces on the turbulence quantities is the dominant effect and the Nusselt number is, therefore, decreased below its forced convective value. At the lower Reynolds nUmber, i.e., between B and C i n Fig. 9.22, the direct effect o f t he buoyancy forces on the mean momentum balance becomes the dominant effect and the Nusselt number rises above its forced convective value. The changes are displayed b y t he numerical results shown i n Fig. 9.23. These results were obtained using a more advanced turbulence model than that discussed here. Based on results similar·to those shown i n Fig. 9.23, the conditions under which mixed convection effects are important i n assisting flow over a vertical plate can be • derived. These conditions are indicated i n Fig. 9.24. In forced convective turbulent boundary layer flow over a plate the velocity and temperature distributions are approximately given by: E XA M P L E 9 .5. ~ = [~r and: 1 = 1 - [~r where U l is the free-stream velocity, T l is the free-stream temperature, Tw is the wall temperature, and 8 is the local boundary layer thickness, the thermal and velocity boundary I Nu GrFixed Forced Turbulent Mixed Convection Re F IGURE 9.22 Variation of Nusselt number variation with Reynolds number in turbulent assisting mixe4 convective flow. 462 Introduction to Convective Heat Transfer Analysis Turbulent Forced Convection ~~ lW~ =5.0 x 1011 ________Grx___________________ ~ 1 i i ~ Grx =5.0 x lOtO .... Grx =5.0x109 102 101L-~~~~~~~w-~~~~~~~~--~~~~~~~ ..,.'....... .' .. <.... , ......... ....... ' ........ . ' .......' .' .'.' . ••••••••••• Laminar Forced Convection 101 104 Reynolds Number, Rex 107 F IGURE 9.23 Numerically predicted variation o f Nusselt number variation with Reynolds number in turbulent assisting mixed convective flow over a vertical plate. (Based on results . obtained by Patel K., Armaly B.F., and Chen T.S., ''Transition from Turbulent Natural to Turbulent Forced Convection Adjacent to an Isothermal Vertical Plate", ASME HTD, Vol. 324, pp. 5 1-56, 1996. With permission.) layers being assumed to have the same thickness. Using these relations derive expressions for the ratioI of the turbulent shear stress i n assisting mixed convective flow to the turbulent shear iIi forced convective flow and for the ratio o f the turbulent heat transfer rate in mixed convective flow to the turbulent heat transfer rate in forced convective flow. Solution. In forced convective flow the turbulent shear stress is given by: Tro = pe2 1 ay au aul ay while in mixed convective flow i t is given by: Tr - P _ e2 au I ay [1 + J{3 g aulaylaulayl ]. aTla y ay au 1 Hence: Tr _ Tro - [1 + J{3 g aulaylaulayl a TlaY] '-1 au _ U l Y 7 1 a y- 7' 8· ~, / But: [1 CHAP'I'ER 9: Combined Convection ,463 107 Turbulent Forced Convection 106 ---------------------------------1 Transition Region I I I 1\)'1 ~ lOS - --------------------------Laminar Forced Convection I I i I I I I I I I I I i. ~ '0 ~ Hf ~ t- 103 I I I 102 I I I I I I I Turb. Free C . onvection 101 101 102 103 . Hf lOS 106 107 ' 108 109 1010 1011 1012 1013 GrashofNumber, Grx F IGURE 9.24 Purely forced, purely free, and mixed convective' regions i n assisting flow over a vertical plate. (Based on results obtained by Patel K., Annaly B.P., and Chen T.S., ''Transition from Turbulent Natural to Turbulent Forced Convection Adjacent .to a n Isothet:mal Verticall'late", ASME HTD, Vol. 324, pp. 5 1-56, 1996. With pe~ission.) ., and: ~ ---' '. aT tJy (Tw - Td [~]~-.1 .!. 7 '8 8 /' . 'Therefore: '.2!. = 1 _ 7Jf3g(Tw - Tl)8 [~]7 'TTO' . u~ 6 8 .~ i.e.: .2!. = 1 _ 7 J Grx ~ [~]\.. :. 'TTo . Re2 x 8 x w here Grx and Rex are the local Grashof number and the local Reynolds number, respectively. T he boundary layer thickness is approximately given by: x = ReO. x U sing this gives: 8 0.036 2 .2!. = 1 'TTo . 0.252J Grx [~]. Re2.2 8 x 9 / 464 Introduction to Convective Heat Transfer Analysis In a similar way. because in forced convective flow the turbulent heat transfer rate is given by: while in mixed convective flow it is given by: qT ::;: If3g arJay pcpce21 ay aT [1 + 2 aulaylaulayl aul ay 1 it follows that: · qT = qTO ~ = 1 _ 0.2521 TTO . G rx Re22 8 x [~]7 6 9.10 I NTERNAL M IXED C ONVECTIVE F LOWS The discussion up to this point in this chapter has been concerned with external mixed convective flows. However,"buoyancy forces can also sometimes have a significant influence on internal flows,see [61] to [67]. O ne example is with the internal cooling o f rotor blades in gas turbine engines. I n this application, the blade rotation generates large centrifugal forces that cause high buoyancy forces to exist in the iIiternal cooling channels and these buoyancy forces can cause the heat transfer rates to b e very cUfferent from those that would exist with purely forced convective flow. Mixed convection effects are also often encountered in nuclear reactor cooling problems, pa.I!icularly during shutdown and emergency situations. As with external mixed convection, the influence o f buoyancy forces on the flow depends on the angle that the buoyancy forces makes to the direction o f the forced flow. T he h eat transfer rate also, of course, depends on the duct cross-sectional shape as well as on whether the flow is laminar or turbulent. For laminar flow in a horizontal pipe, the buoyancy forces can cause a secondary flow that can significantly enhance the heat transfer rate. The nature of the buoyancy-induced secondary motion is illustrated in Fig. 9.25. In a horizontal pipe the buoyancy force acts perpendicular to the direction o f the forced flow. As a result, i n a heated pipe, the fluid rises on the outside o f the pipe wall and Horizontal Pipe with Heated Walls F IGURE 9.25 Secondary flow in mixed convection in a horizontal pipe. CHAPTER 9: Combined Convection 4 (;5 desc~nds n ear the vertical center line o f the pipe forming a pair o f counter-rotating spirals that travel down the pipe. This buoyancy-driven secondary flow cause$ the heat transfer rate to increase significantly on the lower surface o f the pipe. There have been many studies o finternal mixed convection, particularly in circular pipes. The conditions under which flow i n a circular pipe can b e assumed to b e purely forced convective, purely free, and mixed convective have heen presented i n graphical form by Metais and Eckert [62], the form o f these graphs being given in Figs. 9.26 and 9.27. Figure 9.26 applies to flow i na vertical pipe while 'Fig. 9 .27 is for flow i n a horizontal pipe. I n these figUres, the mixed convection regime is defined as the conditions for' which the convective heat transfer deviates by more than 10% from either the purely forced or purely free convective value. Qualitatively, the regions shown in these figures are as is to be expected. A t low ReYnolds and high Rayleigh numbers, free convection is the dominant mode. Conversely, at high Reynolds and low Rayleigh numbers, forced convection dominates. In, these figures,the Rayleigh number i s b ased on the tube diameter and U D i s the pipe length-to-diameter ratio. Note that only the boundary between forced and mixed convection is shown i n Fig. 9.27. The results given in Figs. 9.26 and 9.27 are largely based on experimental results. Here, however, the main interest is i n the analysis of internal mixed convective 1 0-2 < P rDIL < 1 10 1 ~~~~~wu~~wu~~~u-~~u-~~~~~~~~ 10 102 103 104 lOS Ra(DIL) 106 107 108 109 - F IGURE 9.26 Regimes for purely forced, purely free, and mixed convection for flow through vertical pipes [62]. 466 Introduction to Convective Heat Transfer Analysis lOS I I" ' -'ll 1 , ",v. . · " I I III ,I I I I II I I II ~ ~ ....... ---- 104 ~ ~ --'~ f t= 103 ~.'. .... . . 1 __ I ..:1 _ t. . . 1 l U . . ." ,. . . 1 1 " :'s ~ t- D. . !..- ~ 102 rT if ) u; . ..1 .... T ·1 • 'M I I 1 0-2 < PrDIL < 1 III I 109 lOS 107 Rayleigh number, Ra F IGURE 9.27 Regimes for purely forced, purely free, and mixed convection for flow through horizontal pipes [62]. Plow Flow Pipe Plane Duct F IGURE 9.28 Flow in a plane duct and a pipe. flows. A discussion of the equations that have been used in most analysis of internal flows will therefore now be presented. Attention will be restricted to flow in a plane duct and flow in a pipe, these being shown in Fig. 9.28. 9.11 FULLY D EVELOPED MIXED CONVECTIVE F LOW I N A VERTICAL PLANE CHANNEL In order to illustrate some of the characteristics of internal mixed convective flows, consider internal mixed convection between two parallel plates held at different unifonn temperatures as shown in Fig. 9.29. / " CHAPTER 9: Combined Convection 467 t W Velocity and Temperature Profiles N ot Changing in this I Mction y ttttt Flow Plane Duct F IGURE 9.29 Flow in a differentially heated plane duct. I n a sufficiently long channel, the velocity and temperature profiles Will cease to change with distance along the channel, i.e., a fully developed flow will exist, see [68] to [91]. Defining: 8= T wl - T - To , U To = ~, Y = L Urn W (9.102) where Urn is the mean velocity across the duct, To is a reference temperature typically taken as the fluid temperature at the inlet to the channel, and T wi is the temperature of one of the walls of the channel; then in fully developed flow: U = F(Y), 8 = G(Y) (9.103) with F and G being fupctions that do not vary with distance, z, along the duct. Because the mass flow rate is constant, the fact that F is not changing with z implies that the value of U at any y is not changing with z. As a consequence of this it follows that in fully developed flow, v is everywhere 0 which means that the pressure will be the same eveJYwhere across the flow. Further, since the function G is not varying with z and since the wall temperatures T wi and T w2 are also not varying with Z, it follows that the value of T at any y is not changing with z. . Using the above results, it follows that the equations governing the flow are, using the Buossinesq approximation and ignoring viscous dissipation: o= o= dp dz (- + #L ay2 ± {3gp(T - To) a2 u k) p Cp a2T.I.e., a2 T = 0 2' 2 vy vy . .!I .!I (9.105) The pressure, p, is thus being measured relative to that which would exist at the same elevation in the stagnant fluid i f it were at a uniform temperature of To. The positive sign in front of the buoyancy t em applies to buoyancy-assisted flow and the negative sign again applies to buoyancy-opposed flow. 468 Introduction to Convective Heat Transfer Analysis The boundary conditions on these equations are: When y = 0, u = 0 When y = W; u = 0 When y = 0, T = TwI When y = W; T = Tw2 Eqs. (9.104) and (9.105) can be written as: (9.106) o= and: dP - dZ + a2 u Gr a y2 ± Re (J (9.107) (9.108) where: z p Z=--P=ReW' pur Gr = (9.109) V f3g(Tw l - To)W3, Re = umW v2 In terms of the dimensionless variables introduced above, the boundary cOJiditions on the above two equations are: When Y = 0, U = 0 When Y = 1, U = 0 When Y = 0, (J == 1 W henY = 1,(J = r r where: rr = (9.110) Tw2 - To Twl - ,To Eq. (9~108) indicates that the dimensionless temperature varies linearly across the duct. Integrating Eq. (9.108) and applying the, boundary conditions gi.ves: (J = 1 + ( rr - 1)Y (9.111)1 This equation can alternatively be written in more conventional form as: T - To = Twl - To 1+ [Twl - To - 1] Y T To w2 - i.e.: T - Twl = y Tw2 - Twl (9.112) / CHAPTER 9: Combined Convection 469 This is the same temperature distribution as that which would exist with pure conduction across the channel and with purely forced convection, i.e., the buoyancy forces have no effect on the temperature distribution and thus on the heat. transfer rate in the situation considered. Substituting Eq. (9.111) into the momentum equation, Eq. (9.107) gives: .0 = -d Z i.e.: dP + -a u ± - [1 + (rT -1)Y] Gr 2 a y2 R e··· . a2 u af2 = dZ dP Rer [1 +·(rT G =+= I)Y] (9.113) Integrating this equation gives: U= ~; (~)+ ;;[; + (TT -I)~]+ e,Y + C2 e, = - ~: (4 (9.114>.. where C l and C2 are constants of integration. Applying the boundary conditions gives: e2 = 0, )+ ;; [~ + (TT - I)~] (9.115) • Substituting Eq. (9.115) into Eq. (9.114) then gives: U= i.e.: ~ (~)+ ;; [~ + TT-')~] - ~ (~)+ ;; [~+ (TT -I)~]Y ( U = dP (Y - Y2)_ Rer [(Y2_;,Y) + (rT-l) (Y3_ Y)~~ G2 dZ 2 + 6 (9.116) But, because: it follows that: fal U d Y = 1 Substituting Eq. (9.116) into this equation then gives: -~ (I;) + ; ; [ - I; j .e.: (TT ~ 1) ~ ] - 1 . (9.117) _dP = 12 - Re [1 + (1'7' 2- 1)] Gr dZ + 470 Introduction to Convective Heat Transfer Analysis S ubstituting Eq. (9.117) back into Eq. (9.116) then gives: . ~ Gr U = { 1 2 + Re [ 1 + (rr - 1)2 i.e.: I]} (Y - Y2) . 2. Gr ± Re [(y22 · + (rr - 1) (Y - Y) + 2 y3) 3 6 Y)~ ~ U = 6(Y - y2) ± ~; (rr - 1) (112 ) (Y - 3y2 (9.118) I f t he buoyancy forces are negligible, i.e., i f GrlRe =0, the velocity distribution is given by: U = 6(Y - y2) with t he m aximum dimensionless velocity occurring at Y = 0.5 and being equal to 1.5. I t w ill be seen from Eq. (9.118) that the effect o f the buoyancy forces on the velocity profile is characterized b y t he parameter: Gr (rT _ 1) = {3g(Twl - To)W 31v2 [TW2 - To - 1] Re umWlv Twl - To T his buoyancy force effect parameter can be written as: {3g(Twl - Tw 31v2 = GrT 2)W (umWlv) Re where: (9.119) (9.120) is the Grashof number based on Tw l - Tw2: T he reference temperature, To, therefore has no effect on the form o f the velocity profile, i.e., as is to b e expected, i t is only the buoyancy forces that arise due to temperature differences across the flow that have an effect on the velocity profile which can be written as: U ~ 6CY - y2) ± ~: U2)CY - 3y2 + 2y3) (9.121) T he variation o f U with Y for various values o f GrTIRe is shown in Fig. 9.30. Results are only given in this figure for assisting flow, i.e., for the - sign on the buoyancy term in Eq. (9.121). I t w ill be seen from Fig. 9 .30 t hat the buoyancy forces increase the velocity near the hotter wall (at Y = 1). Since the total mass flow rate is fixed, the increase i n velocity near the hot wall is associated with a decrease i n velocity near the cooler wall (at Y = 0). As the parameter Grr/Re increases, the velocity profiles become increasingly distorted and at high values o f GrTIRe flow reversal can occ:ur a djacent to the cooler wall, i.e., a downward flow can occur.near the cooler wall. T he condition under which such a reverse flow occurs can be dedu<:;ed b y considering the shear CHAPTER 9: Combined Convection 471. 1.5 u 0.5 0.0 -0.5~----------------~----------------~ QO ~ 1~ Y F IGURE 9.30 Velocity profiles in fully developed mixed convective flow i n a vertical plane channel. Results are for assisting flow. stress at Y = O. Now the shear stress on the cold wall is given by: TC = fJ- ay au I y =W = W- ay fJ-u m au I Y =l Using Eq. (9.121), this gives: pu'!n TC umW = 6 (1- 2Y) v .' =+= GrT (~)(1- 6Y + 6 y2 )1 Re 12 Y =l = 6 =+= (~)GtT 12 Re (9.122) This equation indicates that in assisting flow this shear stress will be zero when: GrT = 72 Re (9.123) i.e., flow rever~al will occur at the cooler wall if GrTIRe > 72. The above results were for assisting flow, i.e., where the buoyancy forces act in the direction o f the forced flow through the duct. h i opposing flow, the same results are obtained except that the velocity decreases, and ultimately flow reversal, occurs at Y = 1 instead of at Y = O. Air flows vertically upward at a mean velocity of 1.5 mls through a vertical plane channel in which the distance between the the two walls is 3 cm~ The air enters the channel at a temperature of IDoC. The walls of the channel are at temperatures E XAMPLE 9 .6. / 472 Introduction to Convective Heat Transfer Analysis o f 2 0°C a nd 40°C. Assuming fully developed flow, plot the velocity and temperature distribution i n the pipe and find the pressure gradient in the channel. Solution. T he mean air temperature o f the air i n the flow is (20 + 40)/2 temperature for air at standard ambient pressure: = 30°C. A t this Hence: . G rT = ~8(Twl - Tw2)~ Urn" = (11303) X 9.81 X (40 - 20! X 0.03 2 = 24.26 R e. 1.5 X 16.01 X 10- The velocity profile is then given b y Eq. (9.121) U = 6 (Y - y2) - . . 24.26(~)(Y:"" 3y2 + 2y3) 12 y U where: Y = 0 .03' U = I.S where y i s i n m and U is in mls and y i s measured from the cooler wall. T he temperature distribution is given by Eq. (9.112) as: T - Twl Tw2 - Twl i.e.: =Y T -20 4 0-20 i.e.: =Y T = 2 0 + 20Y where T i s i n °C. T he variations o f U a nd T with y as given by these equations are shown in Figs. E 9.6aand E9.6b. T he pressure gradient in the fully developed flow is given by Eq. (9.117) as: _ dP dZ But: Gr = 12 - Gr [1 + Re + (rT - 1)] f = ~8(Twl - To)W 2 Urn" = (11303) X 9.81 X (20 - 10) X 0.032 = 12.13 Re and: . 1.5 X 16.01 X 10-6 r T= Tw2 - To 40 - 10 = . =3 Twl - To 2 0 - 10 H ence, for t he situation being considered: .., (I \.: _ _ _ _ ____ ~ ~,,=_._ _ ___ '_ _ .. ~'~' ___ "_" _ ____"__ - ----.------------- ""-,-··~ __ i_j-~~·----. - -- CHAPI'BR 9: Combined Convection . 473 3 ,.-----,-------.--------. 2 u-ni/s . 1 o~------~-----~------~ 0.00 0.01 '. 0.02 . om y-m 50r-------,--------T------~ FIGURE E9.6a 20 10~-- 0.00 ____ _ _____ _ __ 0.01 0.02 0.03 . FIGURE E9.6b ~ ~~ ~ y -m Therefore, the pressure i s increasing With distance along the duct as a result of the buoyancy forces. But: . Z=~P=.l!.... R eW' put Le.: Z and: = UmW 2lv. z = z x 16.01 x 1~ .;.., 0.01186z 1.5 x 0.03 . - 6. p = 1.164~ 1.52 = O.38l~p . 474 Introduction to Convective Heat Transfer Analysis Hence: d P = 32.19dp dZ dz Using the above results then gives: ~~ = +0.380 Palm Therefore, the pressure gradient is +0.3809 Palm, 9.12 M IXED CONVECTIVE F LOW I N A HORIZONTAL D UCT With mixed convective flow in a horizontal pipe the buoyancy forces act at right angles to the direction of forced flow leading to the generation of a secondary motion as discussed earlier. The equations governing this type of flow will be briefly discussed in this section. Attention will be restricted to fully developed flow, i.e., to flow in which all the flow variables except temperature are not changing with distance, z, along the pipe. I t will also be assumed that the wall heat flux is axially constant and the wall temperature is constant around the periphery although it of course varies with axial distance. Using the coordinate system shown i n Fig. 9.31, the equations governing the flow are, i f the Boussinesq approximation,is adopted and i f viscous dissipation . . . Buoyancy Force.B p,B sint/J Bcost/J b .' ! I t /J//' j F IGURE 9.31 Coordinate system used. CHAPI'ER 9: Combined Convection 475 is neglected: 1 iJ aw - -(vr) + - = 0 'r ar acf> (9.124) viJv + w iJv _ w 2 = _.!. ap + v (a 2v + ! iJv + ~ a2v _ ~ + ~ a w) ar r iJcf> r , , P ar ar2 r, ar r2 a~ r2 r2 acp - f3g(Tw "'" T)coscf> (9.125) : aw ' w aw vw ' V ar + acp - r - r - 1 ap (a 2w 1 ow 1 a2w w 2 a v) pr af/l v ar2 +;: ar + r2 a~ - r2 + r2 iJcp T) sin f/I + f3 geTw - (9.126) (9.127) (9.128) a u, w au 1 ap , (a2ii 1 iJu 1 a2u) v ar + r af/l = - PiJz + v ar2 + ;: iJr + r2 acp2 . u aT + v iJT + w aT iJz ar r aq, = (.!.-)(a 2T + ! aT + ~ a2T) Pr or2 r ar r2 aqil These equations express conservation of mass, conservation of momentum in the r ; cp, and z directions, and conservation of energy, respectively. The referen~e temperature, Tw, is taken as the temperature of the wall o f the pipe at the particular value of z being considered. The pressure can be expressed as: p= P(z) + F (r, f/I) and Eqs. {9.125), (9.126), and (9.127) can then be written as: iJv w iJv w2 v ar + iJf/I,- r r = -p 1 aF (iJ 2V ar v ar2 + +;: ar + r2 iJ~ - r2 + r2 af/l 1 av 1 aZv v 2 aw) (9.129) - f3g(Tw - T)cosf/l iJw w ow vw 1 iJF (a 2w 1 ow 1 a2w w 2 a v) v ar +, r at/J - r = - pr iJq, + v ar2 + ;: ar + r2 a~- 'r2 + r2 iJcp , + f3g(Tw' - T)sit.J.f/I v au + w au = ar r af/l (9.130) -:-.!. dP + v (02u + +! au + ~ a21i.) p dz (I ar2 r ar r2 a~ (9.131) A dimensionless temperature defined as follows is next introduced: "(9.132) T w-Tm w1:tere Tm .is the mean temperature at the particular value, of z being considered. aecause fully develo~ flow is being considered, the temperatur~ Profile, ~ pot be changing with distance along the p~pe, i . e . : : ,,' (I = function (r, f/I) , ' = Tw - T . , ' 476 Introduction to Convective Heat Transfer Analysis The energy equation can therefore be written as: dTw d ao w ao u dz - 0 dz (Tw - Tm) + v ar + r acp (v) (a o 1 ao 1 a 0) = Pr ar2 + r ar + r2 a¢2 2 2 (9.133) But because a uniform axial heat flux is being considered and because: Jo Jo i t follows that T w is increasing linearly with z and that T w energy equation therefore can be writt~n as: D2 qw '1T4 = pCp (R (2'1T ruT dcpdr . Jo J o . = p cp(R (271' ru[Tw - O(Tw - Tm)]dcf>dr T m is a constant. The 2 d Tw ae w ae u dz + v ar + r a4> (v) (a 28 1 ae 1 a o) = Pr ar2 + r ar. + r2 a4>2 (9.134) and the r- and 4>-momentum equations can be written as: av w av w2 1 of (o2v 1 av 1 a2 v v 2 a w) v ar + a4> - r = - p ar + P ar2 + r ar + r2 acf>2 - r2 + r2 acf> r - f3 geT1'.1 - Tm)() cos cf> (9.135) and: aw w ow vw 1 of (o2w l ' oW 1 a2 w . w 2 . av ) v ar + r a4> - -,:- = - pr a4> + v ar2 + r .ar + r2 a¢2 - r2 + r2 a4> + f3 geTw In addition, cons~rvation - Tm)() sin cf> (9.13.6) of total.mass requires that: 2 d dz JrR Jr '1T rudcf>dr oo ~ . 0 (9.137) Eqs. (9.124), (9.135), (9.136), (9.131), (9.137), and (9.134) can be simultaneously solved t o give the solution. For example, the effect of the buoyancy forces can initially be ignored, i.e., v, w, and F can be set equal to zero, Eqs. (9.137) and (9.131) can be solved to give the variation'of u with r, and Eq. (9.134) can be used to give the variation of e with r. These solutions are, o f course, the same as those for forced convective flow in a pipe that were discussed in Cltapter 4. These solutions can then be used in Eqs. (9.135) and (9.136) together with Eq. (9.124) to solve for first approximations for the variations of v, w, and F with r and <p. These results can then be used in Eqs. (9.137) and (9.131) to solve for a second approximation to the variation of u with r and 4> and in Eq. (9.134) to give a second approximation to the variation o f 8 with r and <p. These second approximations to the variations of u and 8' can then be used in Eqs. (9.135), (9.136), and (9.124) to solve for second.approximations to the variations of v, w, and F with r and <p. The procedure can be repeated until convergence to an acceptable degree is achieved. Studies o ffiow in horizontal ducts are discussed m[96] to [112]. J / CHAPTER 9: .Combined Convection 477 9.13 CONCLUDING REMARKS This chapter has been concerned with flows in which the buoyancy forces that ariSe due to the temperature difference have an influence on the flow and heat transfer values despite the presence o f a forced velocity. I n external flows i t was shown that the deviation o f the heat transfer rate from that which would exiSt in purely forced convection was -dependent on the ratio o f the Grashof number to the squat¢ o f the Reynolds number. I t was also shown that i n such flows the Nusselt number can often b e expressed in terms o f the Nusselt numbers that would exist under the same conditions i n purely forced and purely free convective flows. I t was, also show!! that i n turbulent flows, the buoyancy forces can affect the turbulence structure as well as the momentum balance and that in turbulent flows the heat transfer rate can b e - decreased by the buoyancy forces in assisting ,flows whereas i n laminar flows the buoyancy forces essentially always increase the heat transfer rate i n assisting flow. Some consideration was also given to the effect o f buoyancy forces on internal flows. P ROBLEMS 9.1. Water at a temperature of 600 P flows horizontally at a velocity of 8 ftlsec perpendicular to the axis of a I-inch diameter horizontal tube that is kept at a uniform surface temperature of SO°F. Determine whether the buoyancy forces have an effect on the heat transfer rate. 9.2. Consider flow over a wide 20-cm high vertical plate which is held at a uniform surface temperature of 40°C placed in a ir at a temperature of 10°C. I f a vertical forced air flow is induced over the pbite, find the air veloCity at which the forced flow starts to affect the mean he~t transfer rate from the plate. 9.3. A solution that was accurate to first order in the buoyancy parameter.Ox. for near-forced convective laminar two-dimensional boundary layer flow over an isothermal vertical plate was discussed in this chapter. Derive the equations thatwouldaUow a solution that -Was second order accurate in Gx to be obtained. Clearly state the bouiidary conditions on the solution. 9.4. A solution that was accurate to first order in lIG~5 for near free convective laminar twodimensional boundary layer flow over an isothemlal vertical plate was discussed in this chapter. Derive the equations that would allow a solution that was second order accu-/ rate i n I/G~·5, i.e., accurate to order (lIG~·S)2 to be obtained. Clearly state the boundary conditions on the solution. 9.5. A ir at a temperature of lOoC flows upward at a velocity of O.S mls over a wide vertical IS-cm h igh fiat plate which is -maintained at a -uniform surface temperature -of 500C. Plot the variation of the local heat transfer rate with distance along the plate from the leading edge. Also show the variations that would exist in purely forced lmd pU:rely free convective flow. . 9.6. Consider w~ter flow over a vertical flat plate with a height of 30 cm and at a velocity that is such that the Reynolds number based on the height of tlle plate is 5 x lQ4.1f the water / 478 Introduction to Convective Heat Transfer Analysis temperature is 15°C and i f the plate temperature is 35°C, can the effect of the buoyancy forces on the mean heat transfer rate be neglected? W hat is the mean heat transfer rate from the plate? 9.7. The side of a small laboratory fumace.can be idealized as a vertical plate 0.6 m high and ·2.5 m wide. The furnace sides are at 40°C and the surrounding air is at 25°C. I f air i s blown vertically over the side. of the furnace, eStimate the lowest forced a ir ·v~lo_city that would cause the heat-transfer coefficient to depart noticeably from its natw:al convection value. . . -9~8. A ir at a temperature of 30°C flows at a velocity of 1 mls vertically downwards over a Wide vertical flat plate which is held at a uniform surface temperature of 10°C. Plot the variations of the velocity and temperature in the boundary layer at a distance o f 20 em from the leading edge of the plate. Also plot the variations that would exist i f the buoyancy force effects were negligible. 9.9. at a temperature of looe flows. upward over a 0.25 m htgh vertical plate whi~h is kept at a unifonn sUrface temperature of 40oe. Plot the variation of the ·velocity boundary layer thickness and local h eat transfer rate along the plate for air velocities of between 0.2 and 1.5 mls. Assume two-dimensional flow. # - - 9.10. Air at a temperature of lOoe flows vertically upwards over a 0.1 m high vertical plate whose suqace temper~ increases linearly ~m lOoe to 4Q°C with distance along the plate. Numerically determine how the heat transfer rate varies along the plate for various forced velocities between that which gives effectively forced convection and ' that which gives effectively free convection. Assume two-dimensional flow. 9.11. Consider mixed convective laminar boundary layer flow over a horizontal flat plate that is heated to a uniform surface temperature. ID.such a flow there will be a pressure changea.cross the boundary induced by the buoyancy forces, Le.: - iJp = (Jgp(T.".. T.r.) iJy where T<» is the uniform temperature outside the boundary layer. Integrating this equa·tion gives, besause the pressure is expressed relative to the uniform pressure outside the boundary"layer: . P = Jo f6 f3gp(T - T",,)dy - Jo e f3gp(~ - T",,)dy . Using this equation, derive the boundary layer momentum integral eqUation for this type of flow. 9.12. Consider two-dimensional air flow over a square cylinder. How does the Nu.sselt num, ber.vary with Reynolds number at a fixed Grashof number of 10,0001 9.13. _ ir a ta temperature of 10°C flows over a 30-mm diameter horizontal cylinder which A has a uniform surface temperature of 50°C. The a ir flow is at right angles to the axis of the cylinder and in the vertically upwards direction. Determine the air velocity above :. which the flow can be assumed to b e purely forced convective and below which i t can . .. beassum.ed to be-purely free convective. CHAPTER 9: Combined Convection 479 9.14. Consider assisting air flow normal to the axis of a horizontal circular cylinder. Using the equation of the form: Nun = NUN + Nup that was discussed in this chapter, together with standard equations for NUN and NUF, derive equations that define when the flow can be assumed to be purely forced convective and when it can be assumed to be purely free convective. Using these equations, plot the variations of Grashof number with Reynolds number that define these limits. 9.15. In forced convective turbulent boundary layer flow over a plate the velocity and temperature distributions are approximately given by: ~ = [~r and: 1 Where U l is tJ:Ie free-stream velocity, T l is the free-stream temperature, Tw is the wall temperature, and S is the local boundary layer thickness, the thermal and velocity boundary layers being assumed to have the same thickness, and n being a constant with a value of approximately between 5 and 7. Derive expressions for the ratio of the turbulent shear stress in assisting mixed convective flow to the turbulent shear in forced convective flow and for the ratio of the turbulent heat transfer rate in mixed convective flow to the turbulent heat transfer rate in forced convective flow. 9.16. The turbulent kinetic energy equation was derived in Chapter 5 using the momentum equations and assuming buoyancy force effects were negligible. Re-derive this equation starting with momentum equations in which the buoyancy terms are retained. Assume a V'ertica.lly upward flow and use the Boussinesq approximation. 9.17. A numerical procedure for C'alculating the heat transfer rate with turbulent boundary layer flow was discussed in Chapter 5. This procedure used a mixing length-based turbulen~e .model. Discuss the modifications that must be made to this procedure to apply it to mixed convective flow over a verticai plate. 9.18. Air at mean temperature of 40°C flows through ahorizontal pipe that is 1.8 m long with a diameter of 25 mm. The air velocity is such that the Reynolds number is 150. I f the wall of the pipe is kept at a uniform temperature of 100°C determine if the flow can bel assumed to be purely forced convective. . 9.19. Air at a mean temperature of 60°C flows vertically upward through a 20-cm diameter pipe that is 5 m long. The wall of the pipe is kept at a uniform temperature of 30°C. Estimate the flow velocity at which buoyancy force effects will become important. 9.20. Air flows vertically upward at a mean velocity of 1 mls through a vertical plane channel whose walls are temperatures of 30°C and 40°C, the distance between' the walls being 4 cm. The air enters the channel at a temperature of 15°C. Plot the velocity and temperature distribution in the channel assuming fuUy developed flow. . 480 Introduction to Convective Heat Transfer Analysis 9.21. 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